Combining the above with Lemma  and Theorem 4.1 of [1]},
we conclude that in the strict turnstile model
\(\overrightarrow{\mathbf {RSC}}_{1/3}\left(  \textsc {T} _\varepsilon \right)= {\Omega \left( {\varepsilon ^{-2} \log (N) \log \log (mM)} \right)}\) .
Thus we complete the proof.
